Post-tensioned slabs

How an unbonded post-tensioned flat slab strip is designed with the STATIX PT slab calculator on its Eurocode 2 route. It covers the tendon profile and load balancing, the losses of 5.10, the stresses at transfer and in service, ultimate bending with unbonded tendons and punching with the prestress. It also says which checks follow EN 1992-1-1 and which follow ACI 318-19.

1. Learning outcomes

After this lesson you can:

lay out a parabolic tendon profile and find the load it balances;
follow the prestress from the jacking force to the mean force P_m,t with the losses of 5.10.5 and 5.10.6;
check the concrete stresses at transfer and in service (5.10.2.2, 5.10.9, 7.1, 7.2);
check ultimate bending with unbonded tendons (5.10.8) and the minimum bonded steel of 9.2.1.1(1) and (4);
check punching with the prestress (6.4.4(1) σ_cp, 6.4.3(9) V_pd) and detail the slab (9.4, 8.10);
say which checks on the page follow EN 1992-1-1, which follow ACI 318-19, and which you do by hand.

2. Before you start

Lesson 1 (flat slabs: frames, strips and punching to 6.4).
RC Design 1 (Eurocode): bending (lesson 1) and punching (lesson 7).
The PT slab calculator, set to Eurocode 2 (EN 1992-1-1) + ACI 318-19 under More inputs.
A pocket calculator and about 75 minutes.
The same lesson on the SA code: RC Design 2, lesson 2 (SANS 10100-1 + ACI 318-19).

What the PT slab page’s Eurocode 2 route uses.
Strip and analysis. One strip of the full panel width l₂, analysed as a continuous beam on its columns. The columns above and below act as springs 4EI/H, fixed at their far ends. The section is the gross section, with E_cm of Table 3.1. The imposed load goes on all, odd and even spans.
Tendons and profile. Unbonded 7-wire monostrands, anchored at mid-depth at both ends. Each span is made of parabolas. Over each interior column a reverse curve runs to an inflection point 0.1 of the span from the column on each side (a practice value). On this route the concrete strength boxes are labelled f_ck and f_ck(t).
Losses. Jacking at 0.75 f_pk, checked by 5.10.2.1 and 5.10.3(2) with f_p0,1k = 0.88 f_pk. Friction with μ 0.05 and a wobble of 0.0025 per metre, and a wedge draw-in of 6 mm: values for greased strand in plastic sheaths. EN 1992-1-1 takes these from the system’s European Technical Approval, with fallbacks in 5.10.5.2(2) and (3) (section 4). The long-term losses follow eq. 5.46, with creep from Annex B.1 (stressed at 7 days, cement class N), the final shrinkage of 3.1.4(6) and relaxation class 2 at 500 000 hours (3.3.2(7), (8)). All are mean values along the tendon (5.10.6(3)).
Stresses. The prestress in the stress checks is taken times 0.95 and 1.05 (5.10.9).
ULS. EN 1990 6.10, 1.35 G + 1.5 Q, with the secondary moment at 1.0. Bending with α_cc 1.0 (the recommended value of 3.1.6(1); the other pages of this course use 0.85) and Δσ_p,ULS 100 MPa (5.10.8(2)). Bonded bars at f_yk 450 MPa and 25 mm cover.
Punching. 6.4 with the β values of 6.4.3(6), v_Rd,c + 0.1 σ_cp, and V_pd taken as 0. The effective depths come from the bars the page places: with Y16 over the columns, d 209 mm for punching and d_s 217 mm in bending.
Deflection, self-weight and frequency. Deflection with E_c,eff = E_cm/(1 + φ). Self-weight at 24 kN/m³. A floor frequency row, labelled SANS 10160-1 8.2.5 on this route: the same rule as EN 1990 A1.4.4(2), so the limit is yours to give.
Some ACI 318-19 flat-slab rules and a few SANS rules stay on this route: section 7 lists them. Other annexes may use other values.

3. The idea

Why post-tension a slab. A tendon draped low in the spans and high over the columns, then stressed, pushes up on the concrete along its curve. Chosen well, that upward load balances most of the self-weight, so a thinner slab spans further with little deflection and few cracks, and the concrete is squeezed by the tendon force as well.

Unbonded monostrands. Most post-tensioned flat slabs use single greased strands in plastic sheaths. They are not bonded to the concrete, so the strand force reaches the slab only at the anchors and along the curves, and the strain in the strand is spread along its whole length. EN 1992-1-1 treats such a slab like reinforced concrete for cracking (7.3.1(6)) and gives special rules for its ultimate strength (5.10.8) and its minimum steel (9.2.1.1(4)). The cover to an unbonded tendon follows the system’s approval (4.4.1.2(10)).

The tendon profile. Each span is made of parabolas: high over the interior columns, low near mid-span, with a short reverse curve over each column so the tendon has no kink. The drape a sets how much it can lift. The calculator lays the profile out from the covers you give to the strand centre. It puts the inflection point of the reverse curve 0.1 of the span from the column on each side, a practice value: the calc sheet states it as “inflection at 0.1 L”; the page does not flag it.

Load balancing. A parabolic tendon with force P and drape a over a span L pushes up with 8Pa/L². The number of strands per metre is chosen so that this balances most of the self-weight. An end span anchored at mid-depth has less drape than an interior span, so it balances less over the same length; designers make the end spans shorter, or add strands there. The stress, strength and deflection checks then decide.

Equivalent loads and the secondary moment. The calculator replaces the tendon by the loads it puts on the concrete (the upward and downward curve loads and the anchor forces) and analyses the strip under them at the mean force. That gives the whole prestress moment M_PT: the primary part P e from the tendon’s eccentricity, plus a secondary part M₂ because the columns and the continuity stop the slab from bending freely (5.10.7(2)). The stress checks use all of M_PT; at the ULS the tendon is part of the resistance and only M₂ is an action, with γ_P 1.0 (2.4.2.2(1), 5.10.8(1)).

4. Prestress force and losses (5.10)

The force along the tendon. The jack pulls the strand to P_max. Friction on the curves and on small unintended wiggles lowers the force away from the jack: the force left is P_max e^{−μ(θ + kx)}, from the loss of eq. 5.45 (5.10.5.2). When the wedges seat, the strand slips back a few millimetres and the force near the jack drops, over a length the slip reaches (5.10.5.3). Stressing one strand after another shortens the concrete under the strands already locked (5.10.5.1). What is left is P_m0. Over the years creep, shrinkage and relaxation take P_m0 to P_m,t (5.10.6). For an unbonded tendon the code works with mean values along the whole tendon (5.10.6(3)); the calculator uses the mean force everywhere.

How the page takes each loss. EN 1992-1-1 gives the formulae and takes the friction coefficients and the draw-in from the system’s European Technical Approval (5.10.5.2, 5.10.5.3). Without an approval it gives fallbacks: μ from Table 5.1 (5.10.5.2(2)) and k between 0.005 and 0.01 per metre (5.10.5.2(3)). Table 5.1 has no column for an internal unbonded strand in a greased sheath. The page writes the wobble term as K x with K = μk = 0.0025 per metre: with its μ 0.05 that is k = 0.05 per metre, well above the fallback range, so the page’s wobble loss is on the high side. Use the values in your supplier’s approval for all three. The page takes the whole final shrinkage, with none of it spent before stressing, which is on the safe side.

Jacking limits. 5.10.2.1 limits the stress at the jack while stressing; 5.10.3(2) limits the stress left in the tendon once friction, draw-in and elastic shortening have acted. The page jacks every strand to 0.75 f_pk and checks both: the second against the largest force anywhere along the tendon after the immediate losses, which lies where the draw-in stops.

The long-term losses together. Eq. 5.46 adds the shrinkage, the relaxation (cut to 0.8 because the concrete shortens as the steel relaxes) and the creep under the concrete stress next to the tendon, and divides by a term for the concrete that holds the tendon back. The page takes the concrete stress as the mean precompression after the immediate losses, with the loads left out, and z_cp as 0. Both give a slightly larger loss.

5. Stresses at transfer and in service

Stress limits. The slab is checked twice: at transfer, when the prestress is highest and only the self-weight acts, and in service with all the loads. 5.10.9 asks for two values of the prestress, 0.95 and 1.05 times the mean for unbonded tendons, and the page keeps whichever is worse at each point. The class is the page’s name for the tension you allow: class 2 keeps the slab uncracked, tension at most f_ctm (7.1(2)); class 1 allows none. 7.2(2) asks for 0.6 f_ck in exposure classes XD, XF and XS; the page applies it in every class, on the safe side.

Working out a stress. On the uncracked gross section σ = P/A ± M/Z, with P the tendon force times 0.95 or 1.05 and M the moment from the loads plus the whole prestress moment M_PT. Over a column the hogging moment takes compression off the top. Where the tension stays below f_ctm under the characteristic combination, 7.3.2(4) asks for no minimum crack steel; above it the slab cracks and is treated as reinforced concrete (7.3.1(6)), and the page tells you to deepen the slab or add strands, as it does not check crack widths.

6. Strength, minimum steel and punching

Ultimate bending with unbonded tendons. An unbonded strand is not strained with the concrete next to it, so it gains little stress as the slab fails. Without a calculation over the whole member, 5.10.8(2) lets you take the effective stress plus Δσ_p,ULS, 100 MPa recommended, capped here at the design strength f_p0,1k/1.15. The design moment includes the secondary moment at γ_P 1.0. The page checks every point along the strip, the column centre lines included, which is on the safe side of the moment at the face of 5.3.2.2(3).

Minimum bonded steel. A slab with only unbonded tendons can fail soon after it cracks. 5.10.1(5)P asks that brittle failure is avoided, and method A of 5.10.1(6) does it with the minimum steel of 9.2.1. For unbonded tendons, 9.2.1.1(4) asks that the bending strength is larger than the cracking moment; 1.15 times the cracking moment is sufficient. Take M_cr as the moment that cracks the slab with the prestress acting, Z(f_ctm + P/A) + P e, comparable with M_Ed. For slabs, 9.3.1.1(1) also brings in 9.2.1.1(1): A_s,min of eq. 9.1N, a reinforced-concrete rule that counts the bonded bars only. Example A checks both. The page checks neither: it sizes its bonded bars by ACI 318-19 8.6.2.3 and by the 0.15 % rule of SANS 10100-1. Check them by hand.

Bonded steel at the columns. EN 1992-1-1 has no rule of its own for the bonded bars over the columns of a post-tensioned flat slab, so the page keeps the ACI 318-19 rule. That rule asks for 0.000 75 times the slab depth times the larger width of the two crossing frames: the panel width, or across it the mean of the adjacent spans (the page takes the end span at an edge column). The bars go in a band c + 3h wide, at least four bars at most 305 mm apart, ln/6 past the face. 9.4.1(3) still asks for at least two bottom bars each way through every internal column; the tendons there are high in the slab and do not count. Place them by hand.

Punching with prestress. 6.4.4(1) adds k₁σ_cp to the resistance, k₁ 0.1 recommended, with σ_cp the mean precompression of the two directions. One strip knows only its own: give the other in “P/A in the other direction”; left blank, the Eurocode route takes 0 and warns you. The steel ratio ρ_l counts only bonded bars (the tendons are unbonded) over the column width plus 3d each side; the page takes it so, with the same ratio both ways (√(ρ_ly ρ_lz)). 6.4.3(9) lets the vertical component V_pd of tendons crossing the control perimeter count in your favour, but 9.4.3(2) counts only tendons within 0.5d of the column; the page takes V_pd as 0. A strip in one direction holds only its own tendons’ prestress reaction in V_Ed, so add the other direction’s before you take off the V_pd of both. When links are needed the page designs them by 6.52 without σ_cp, on the safe side; links need a slab at least 200 mm deep (9.3.2(1)).

Load arrangements. 5.1.3(1) asks for the imposed load on alternate spans and on any two adjacent spans, with γ_GG_k everywhere. The PT page loads all spans, the odd spans and the even spans. On two spans that is the same thing; on three or more, two loaded neighbours give a little more hogging and column load between them than all spans loaded. Check that case with a frame program or by hand.

7. Deflection, detailing and which code

Deflection. With the prestress balancing most of the weight the slab may hardly move, or even lift. The page works the strip deflection on the gross section with the effective modulus of 7.4.3(5) for the long term, and checks span/250 for the final deflection and span/500 for the part after the partitions go in. It keeps the SANS 10100-1 upward limit of span/300 with finishes; EN 1992-1-1 7.4.1(4) limits only the camber set in the formwork. For the final deflection the page adds the rest of the imposed load to the quasi-permanent part, on the safe side. This is the deflection along the column line; the panel centre adds the strip the other way.

Spacing, gaps and fire. EN 1992-1-1 gives no largest spacing for the tendons of a flat slab; the page keeps ACI 318-19 8.7.2.3. The least clear gap between ducts is in Figure 8.15 (8.10.1.3(3)); the page checks the SANS 10100-1 gap of aggregate size + 5 mm, so check 50 mm by hand, taking the sheath as the duct. For fire the page checks the depth and the axis distance of the strands for a flat slab in EN 1992-1-2 Table 5.9, plus 15 mm for strands (5.2(5)).

Anchorage zones. Behind each anchor the tendon force spreads out into the slab, and the spreading pulls the concrete apart across it. 8.10.3 asks for this to be designed by a strut-and-tie model; for a load spreading into a depth or width b from a plate a wide, 6.5.3(3) gives the tie force directly. Use F = 1.2 P_max for local effects. The page keeps the bursting table of SANS 10100-1 on this route; do the EN 1992-1-1 check by hand (Example B). The anchorage device and its spiral are the supplier’s.

Three codes on one route. EN 1992-1-1 covers prestressed concrete but has no rules of its own for the minimum precompression, the tendon spacing or the bonded bars of an unbonded flat slab. The page’s Eurocode 2 route therefore uses EN 1992-1-1 where it speaks, ACI 318-19 for those flat-slab rules, and keeps a few SANS rules; every row on the page names its clause. The frequency row is labelled SANS 10160-1 8.2.5 on this route; the EN rule is the same, EN 1990 A1.4.4(2), with the limit left to you. Outside the rows the page also keeps the self-weight at 24 kN/m³, the full-panel strip cited to SANS 10100-1 4.6.5.1.1 on the calc sheet, and a warning about the SANS duct cover rule (5.9.3.3). The last lines list the EN 1992-1-1 items the page does not check: the examples do them by hand.